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Some questions on the Ethereum zk-SNARKS implementation

OK, so I went through the Goldwasser, Blum, etc. papers tracing things from the beginning (in regards to other things I've long discovered you reach the A-HA! moment quicker when going back to where this or that thing originally popped up from), but I still kind of don't entirely understand how this is supposed to work on Ethereum.

Basically, a trusted setup for whatever goal or purpose generates a private/public key pair, burns the private part and from the public one generates yet another key pair of proving and verifying key.

The verifying part on the blockchain comes as a bunch of precompiled smart contracts, so basically, the blockchain here acts as verifier of proofs of computational integrity? Is that correct?